Partial Differential Equations — Foundations¶

Job: Classify linear second-order PDEs by principal-symbol signature; fix well-posed data classes; distinguish classical vs weak solutions.

Core¶

Linear second-order (two variables, principal part): $$ A u_{xx}+2B u_{xy}+C u_{yy}+\text{lower}=G,\qquad \Delta=B^2-AC. $$ In $n$ variables the principal symbol is the quadratic form $\sigma(x,\xi)=\sum a^{ij}(x)\xi_i\xi_j$; type is the signature of $(a^{ij})$.

Type	Discriminant / signature	Model	Typical data
Elliptic	$\Delta<0$ (definite)	$\Delta u=f$ (Poisson)	Dirichlet/Neumann on $\partial\Omega$
Parabolic	$\Delta=0$ (degenerate + evolution)	$u_t-\Delta u=f$	IC $+ $ lateral BC
Hyperbolic	$\Delta>0$ (indefinite)	$u_{tt}-c^2\Delta u=f$	Cauchy data on spacelike surface

Well-posedness (Hadamard): existence, uniqueness, continuous dependence on data in appropriate topologies. Elliptic BVP on closed domains; pure IVP for Laplace is classically ill-posed.

Classical solution: $u\in C^k$ with $k=\mathrm{order}$, PDE holds pointwise.

Distributional and weak solutions¶

Distributions: $\mathcal{D}'(\Omega)=(C_c^\infty(\Omega))'$. A distribution $u$ solves $Lu=f$ in the distributional sense if $$ \langle u,L^*\varphi\rangle=\langle f,\varphi\rangle\quad\forall\varphi\in C_c^\infty(\Omega), $$ where $L^*$ is the formal adjoint (integrate by parts, no boundary terms because $\varphi$ is compactly supported).

Notion	Meaning
Distributional solution	$Lu=f$ in $\mathcal{D}'$ as above
Weak / variational solution	$u$ in a Sobolev space (e.g. $H_0^1$) satisfying the bilinear form identity $a(u,v)=\langle f,v\rangle$ for all test $v$ in that space
Mild solution	variation of constants / semigroup form (evolution equations)
Viscosity solution	test-function max/min inequalities (fully nonlinear / 1st order)

Weak solutions need not be classical. Elliptic regularity: if $L$ is uniformly elliptic with smooth coefficients and $f\in H^s_{\mathrm{loc}}$, then distributional solutions satisfy $u\in H^{s+2}_{\mathrm{loc}}$ (and bootstrap to smooth if $f$ smooth). Weyl’s lemma: distributional solutions of $\Delta u=0$ are smooth (in fact analytic).

Fundamental solutions. $E\in\mathcal{D}'$ with $LE=\delta$ yields solutions $u=E*f$ for compactly supported $f$ (convolution); singularities of $E$ encode propagation (cf. microlocal analysis).

Energy methods (sketch)¶

Multiply the PDE by a test function (often $u$ or $u_t$), integrate, integrate by parts, absorb lower-order terms.

Model	Energy identity / inequality
Wave $u_{tt}-c^2\Delta u=0$	$\frac{d}{dt}\mathcal{E}=0$ with $\mathcal{E}(t)=\frac12\int(u_t^2+c^2
Heat $u_t-\Delta u=0$	$\frac{d}{dt}\\|u\\|_{L^2}^2=-2\\|\nabla u\\|_{L^2}^2\le 0$; higher $H^k$ estimates by applying $\partial^\alpha$
Coercive elliptic	$a(u,u)\ge \alpha\\|u\\|_{H^1}^2$ $\Rightarrow$ uniqueness + continuous dependence via Lax–Milgram
Nonlinear / quasilinear	a priori $L^2$ or $H^1$ bounds; weak compactness (Alaoglu / Aubin–Lions) to pass to limits

Grönwall upgrades integral inequalities $\frac{d}{dt}E\le C E+F$ to exponential control — standard for uniqueness of energy-class solutions. Hyperbolic energy estimates on spacelike slices give well-posedness for symmetric hyperbolic / strictly hyperbolic systems without constructing fundamental solutions explicitly.

Maximum principles (models): - Heat / elliptic 2nd order: weak and strong maximum principles; Hopf boundary point lemma. - Laplace: mean-value property; harmonic $\Rightarrow$ real-analytic. - Heat kernel: $\Phi(x,t)=(4\pi t)^{-n/2}e^{-|x|^2/(4t)}$; infinite propagation speed for the heat equation.

Method of characteristics (first-order quasilinear $a\cdot\nabla u=b$): ODE transport along $\dot x=a(x,u)$; shocks when projection folds (Burgers).

Separation of variables: product ansätze reduce linear constant-coeff models on product domains to Sturm–Liouville eigenproblems; completeness of eigenfunctions is spectral theory of compact self-adjoint resolvents.

See it¶

Weak form of $-\Delta u=f$ on $\Omega\subset\mathbb{R}^n$ bounded, $u|_{\partial\Omega}=0$: find $u\in H_0^1(\Omega)$ with $$ \int_\Omega\nabla u\cdot\nabla v=\int_\Omega f v\quad\forall v\in H_0^1. $$ Lax–Milgram (coercivity of Dirichlet form via Poincaré) $\Rightarrow$ unique weak solution; $f\in L^2$ and smooth $\partial\Omega$ $\Rightarrow$ $u\in H^2\cap H_0^1$ (shift theorem).

Check¶

Prove or disprove: every weak solution of $\Delta u=0$ on a ball is $C^\infty$.

True in the classical elliptic setting: Weyl's lemma / elliptic regularity for constant-coeff Laplace. Weak harmonic $\Rightarrow$ smooth (in fact analytic). Counterexamples require rough coefficients or lower-order structure that destroys ellipticity.

Why is the Cauchy problem for Laplace typically ill-posed?

High-frequency modes $e^{\pm\xi y}\sin(\xi x)$ grow exponentially in $y$; small $C^k$ perturbations of Cauchy data can produce huge solutions (Hadamard). Hyperbolic principal symbol is required for continuous dependence in standard energy spaces.

Pitfalls¶

Variable-coeff type can change by point (Tricomi $u_{xx}+x u_{yy}=0$); characteristics may not foliate.
Weak $\neq$ viscosity: viscosity solutions (Crandall–Lions) are the right notion for many fully nonlinear first-order / degenerate elliptic equations (HJ, Monge–Ampère).
Distributional solutions ignore boundary conditions — boundary traces need Sobolev trace theorems / weak formulations in $H^1$, not just $\mathcal{D}'$.
Infinite propagation for heat is a modeling idealization; finite-speed parabolic approximations exist (porous medium, hyperbolic heat).
Energy equality may fail for weak solutions of nonlinear equations; often only energy inequality holds.

Refs¶

Evans, Partial Differential Equations; Hörmander, ALPDO I–IV; Gilbarg–Trudinger (elliptic); Lions–Magenes (Sobolev–trace).

Next¶

DE foundations · Microlocal analysis · SPDE · Functional analysis · Fourier analysis · PDE index · Elliptic Theory · Dispersive & Wave · Navier–Stokes · Calculus of Variations · SCV / analytic methods — the Hörmander $L^2$ as elliptic / $L^2$ PDE · Dynamical systems — Navier–Stokes / fluid dynamics as an infinite-dimensional dynamical system

Model	Energy identity / inequality
Wave \(u_{tt}-c^2\Delta u=0\)	\(\frac{d}{dt}\mathcal{E}=0\) with $\mathcal{E}(t)=\frac12\int(u_t^2+c^2
Heat \(u_t-\Delta u=0\)	\(\frac{d}{dt}\\|u\\|_{L^2}^2=-2\\|\nabla u\\|_{L^2}^2\le 0\); higher \(H^k\) estimates by applying \(\partial^\alpha\)
Coercive elliptic	\(a(u,u)\ge \alpha\\|u\\|_{H^1}^2\) \(\Rightarrow\) uniqueness + continuous dependence via Lax–Milgram
Nonlinear / quasilinear	a priori \(L^2\) or \(H^1\) bounds; weak compactness (Alaoglu / Aubin–Lions) to pass to limits